--------------------------------------------------------- Elemental(2f0+1,f0=1) = 100 -- shifted to align with f0 --> 100 Pat(2n+1,n=1) = 100 Size = 3 Number of 1's = 1; 1/3 = 0.333333 Number of 0's = 2; 2/3 = 0.666667 Total Candidates= 1 Singleton Candidates = 0; 0/1 = 0.000000 Twin Candidates = 1; 1/1 = 1.000000 --------------------------------------------------------- --------------------------------------------------------- Elemental(2f0+1,f0=2) = 01000 -- shifted to align with f0 --> 01000 Pat(2n+1,n=2) = 110100100101100 Size = 15 Number of 1's = 7; 7/15 = 0.466667 Number of 0's = 8; 8/15 = 0.533333 Total Candidates= 5 Singleton Candidates = 2; 2/5 = 0.400000 Twin Candidates = 3; 3/5 = 0.600000 --------------------------------------------------------- --------------------------------------------------------- Elemental(2f0+1,f0=3) = 1000000 -- shifted to align with f0 --> 0010000 Pat(2n+1,n=3) = 111100100101100110100101101100110100110101101110100100101110110101100101100110110100101100110100100111100 Size = 105 Number of 1's = 57; 57/105 = 0.542857 Number of 0's = 48; 48/105 = 0.457143 Total Candidates= 33 Singleton Candidates = 18; 18/33 = 0.545455 Twin Candidates = 15; 15/33 = 0.454545 --------------------------------------------------------- --------------------------------------------------------- Elemental(2f0+1,f0=5) = 10000000000 -- shifted to align with f0 --> 00001000000 Pat(2n+1,n=4) = 111110100101100110100101101100110100110101101110100100101111110101100111100110110100101100111100100111110111100100101100110101101101100110100110101101110100101101110110111100101100110110100101100110100110111100111100100101101110100101111100110100110101101111100100101110110101100101100110111100101100110100100111100111100101101100110110101101100110100110101101110100110101110110101100101101110110100111100110100100111100111100100101110110100101101100110101110101101110100100101110110101101101100110110100101100110100100111100111100110101100110100101101101110100110111101110100100101110111101100101110110110100101100110101100111100111100100101100110100101101100110110110101101110100100101110110101110101100110110100101101110100100111100111100100101100111100101101110110100110101101110101100101110110101100101100110110101101100110110100111100111100100101100110100111101100110100110101101110100100111110110101100101100111110100101110110100100111100111101100101100110100101101100110100111101101110110100101110110101100101100110110110101100110100100111101111100100111100110100101101100111100110101111110100100101110110101100101100110110100101100110100101111100 Size = 1155 Number of 1's = 675; 675/1155 = 0.584416 Number of 0's = 480; 480/1155 = 0.415584 Total Candidates= 345 Singleton Candidates = 210; 210/345 = 0.608696 Twin Candidates = 135; 135/345 = 0.391304 --------------------------------------------------------- --------------------------------------------------------- Elemental(2f0+1,f0=6) = 1000000000000 -- shifted to align with f0 --> 0000010000000 Pat(2n+1,n=5) = 11111110010110011010010110110011010011010110111010010010111111010110011110011011010110110011110010011111011111010010110011110110110110011010011010111111010010110111011011110010110011011011010110011010111011110011110010010110111010010111110111010011010110111110010011111011010110110110011011110010110011011010011110011110010110110011011010110111011010011010110111010011010111011010111010110111011110011110011010010011110011110010010111111010010110110011010111011110111010010110111011010110110110011011010010110011110010011110011110011010111011010010110110111010011011110111010011010111011110110010111011011010010110011010110011110111110010010110011010010111110011011011110110111010010010111011011111010110011111010010110111010010011111011110010010110011110010110111011010011010110111010110010111011010110010110011011010110110111011010011110011110010011110011010011110110011010011010110111011010011111011110110010110011111010010111011010010011110011110110010110011010011110110011010111110110111011010010111011010110010110111011011010110011010010011110111110010111110011010010110110011111011010111111110010010111011010110010111011011010010110011010010111110011111011010110011010110110110011010011010110111010010010111111010110011110011011010011110011110010111111011110010010110011011110110110011110011010110111010010110111011011110010110011011010010110011010011011110011110110010110111010010111110011010011010110111110010010111011010110011110011011110110110011010010011110011111010110110011111010110110011010011010111111010011010111011010110010110111011011011110011010110011110011110010010111011010010110110111010111010110111010010011111011010110110110011011010010110011011010011110011110011010110011010010110111111010011011110111010010010111011110111010111011011110010110011010110011110011110010010110111010010110110011011011011110111010010110111011010111010110011011010010110111110010011110011110010010111011110010110111011010011010110111010111010111011010110010110011011010110110011011010011110111110010010110011010011111110011010011110110111010010011111011011110010110011111010010111011010010011111011110110010110011010010110110011010011110110111011110010111011010110010110011011011010110111010010011110111110010011110011010010110110011110011010111111011010010111011110110010110011011010010111011010010111110011111010010110011010011110110011010111010110111010010010111111010110011110111011010010110011110010011111011110010110110011010110110110011011011010110111110010110111011011110010111011011010010110011010011011110011110011010110111010110111110011010011010110111110010010111111010110010110011011110011110011010010111110011110010110110011011010110110011110011010110111010011010111011010110010110111011010011110011010011011110011110110010111011010010110110011010111010110111010010010111011010110111110011011010110110011010010011110011111011010110011110010110110111010011011111111010010010111011110110010111011011011010110011010110011110011110010010110011010010110110111011011010110111010010011111011010111110110011011010010110111011010011110011110010010110011110010110111011010011010110111010110010111011010111010110011011110110110011011010011110011110010010110111010011110110011010011011110111010010111111011010110010110011111010010111011110010011110011110110010111011010010110110011010011110110111011011010111011010110010110011011011010110011010010011110111110010011110011010010111110011110011110111111010010010111011011110010110011111010010110011010010111111011111010010110011010010110110011010011010110111010110010111111010110011110011011010010110111110010011111011110010011110011010110110110011010011010110111011010110111011111110010110011011010010111011010011011110011110010010110111010011111110011010111010110111110010010111011010110010110111011110010110011010010011110011110010110110011011010110110011011011010110111110011010111011010110010111111011010011110011010010011110011110011010111011010110110110011010111010110111010010010111111010110110110011011010011110011010010111110011110011010110011011010110110111110011011110111010010010111011110110010111011011010010110011010111011110011110110010110011010010110110011011011010110111010010010111011010111011110011011010110110111010010011110011111010010110011110010110111011010011010111111010110010111011010110010110011011011110110011011110011110011110010010110011010011110110111010011010110111010010011111011010110110110011111010010111011011010011110011110110010110011010010110111011010011110110111011010010111011010111010110011011111010110011010010011110111110010011110111010010110110011110011011111111010010110111011010110010110011011010010110011110010111110011111010010111011010010110110011010011010110111010011010111111010110011110011011010010110011110010011111111110010010110011010110111110011010011110110111010010110111011011110010110011111010010110011010011011111011110010010110111010010111110011010011010110111110110010111011010110010110011011110010110111010010011110011110010111110011011010110110011010011010110111011011010111011110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110100111111100 Size = 15015 Number of 1's = 9255; 9255/15015 = 0.616384 Number of 0's = 5760; 5760/15015 = 0.383616 Total Candidates= 4275 Singleton Candidates = 2790; 2790/4275 = 0.652632 Twin Candidates = 1485; 1485/4275 = 0.347368 --------------------------------------------------------- --------------------------------------------------------- Elemental(2f0+1,f0=8) = 10000000000000000 -- shifted to align with f0 --> 00000001000000000 Pat(2n+1,n=6) = TOO BIG! Size = 255255 Number of 1's = 163095; 163095/255255 = 0.638949 Number of 0's = 92160; 92160/255255 = 0.361051 Total Candidates= 69885 Singleton Candidates = 47610; 47610/69885 = 0.681262 Twin Candidates = 22275; 22275/69885 = 0.318738 --------------------------------------------------------- --------------------------------------------------------- Elemental(2f0+1,f0=9) = 1000000000000000000 -- shifted to align with f0 --> 0000000010000000000 Pat(2n+1,n=7) = TOO BIG! Size = 4849845 Number of 1's = 3190965; 3190965/4849845 = 0.657952 Number of 0's = 1658880; 1658880/4849845 = 0.342048 Total Candidates= 1280205 Singleton Candidates = 901530; 901530/1280205 = 0.704208 Twin Candidates = 378675; 378675/1280205 = 0.295792 --------------------------------------------------------- --------------------------------------------------------- Elemental(2f0+1,f0=11) = 10000000000000000000000 -- shifted to align with f0 --> 00000000001000000000000 Pat(2n+1,n=8) = TOO BIG! Size = 111546435 Number of 1's = 75051075; 75051075/111546435 = 0.672824 Number of 0's = 36495360; 36495360/111546435 = 0.327176 Total Candidates= 28543185 Singleton Candidates = 20591010; 20591010/28543185 = 0.721398 Twin Candidates = 7952175; 7952175/28543185 = 0.278602 --------------------------------------------------------- IT IS ALSO POSSIBLE TO ALGORITHMICALLY DETERMINE THE # OF ONES AND ZEROS AS OUTLINED ON MY WEBSITE. n=2; Ones->7 Zeros->8 zeros/(ones+zeros) = 0.533333 n=3; Ones->57 Zeros->48 zeros/(ones+zeros) = 0.457143 n=4; Ones->675 Zeros->480 zeros/(ones+zeros) = 0.415584 n=5; Ones->9255 Zeros->5760 zeros/(ones+zeros) = 0.383616 n=6; Ones->163095 Zeros->92160 zeros/(ones+zeros) = 0.361051 n=7; Ones->3190965 Zeros->1658880 zeros/(ones+zeros) = 0.342048 n=8; Ones->75051075 Zeros->36495360 zeros/(ones+zeros) = 0.327176 --------------------------------------------------------- COMPARE THE ABOVE VALUES TO THE NUMBER OF PRIMES/ODD BETWEEN P(n)->P(n)^2 NOTE: P(n)->P(n)^2 compares to n-1 above because (Pat(n-1)) is in effect between P(n)->P(n)^2) P(1)->P(1)^2 : 3->9 numPrime/numOdd = 2/2 = 1.000000 P(2)->P(2)^2 : 5->25 numPrime/numOdd = 6/9 = 0.666667 P(3)->P(3)^2 : 7->49 numPrime/numOdd = 11/20 = 0.550000 P(4)->P(4)^2 : 11->121 numPrime/numOdd = 25/54 = 0.462963 P(5)->P(5)^2 : 13->169 numPrime/numOdd = 33/77 = 0.428571 P(6)->P(6)^2 : 17->289 numPrime/numOdd = 54/135 = 0.400000 P(7)->P(7)^2 : 19->361 numPrime/numOdd = 64/170 = 0.376471 P(8)->P(8)^2 : 23->529 numPrime/numOdd = 90/252 = 0.357143 P(9)->P(9)^2 : 29->841 numPrime/numOdd = 136/405 = 0.335802 P(10)->P(10)^2 : 31->961 numPrime/numOdd = 151/464 = 0.325431 P(11)->P(11)^2 : 37->1369 numPrime/numOdd = 207/665 = 0.311278 P(12)->P(12)^2 : 41->1681 numPrime/numOdd = 250/819 = 0.305250 P(13)->P(13)^2 : 43->1849 numPrime/numOdd = 269/902 = 0.298226 P(14)->P(14)^2 : 47->2209 numPrime/numOdd = 314/1080 = 0.290741 P(15)->P(15)^2 : 53->2809 numPrime/numOdd = 393/1377 = 0.285403 P(16)->P(16)^2 : 59->3481 numPrime/numOdd = 470/1710 = 0.274854 P(17)->P(17)^2 : 61->3721 numPrime/numOdd = 501/1829 = 0.273920 P(18)->P(18)^2 : 67->4489 numPrime/numOdd = 590/2210 = 0.266968 P(19)->P(19)^2 : 71->5041 numPrime/numOdd = 655/2484 = 0.263688 P(20)->P(20)^2 : 73->5329 numPrime/numOdd = 684/2627 = 0.260373 P(21)->P(21)^2 : 79->6241 numPrime/numOdd = 789/3080 = 0.256169 P(22)->P(22)^2 : 83->6889 numPrime/numOdd = 863/3402 = 0.253674 P(23)->P(23)^2 : 89->7921 numPrime/numOdd = 976/3915 = 0.249298 P(24)->P(24)^2 : 97->9409 numPrime/numOdd = 1138/4655 = 0.244468 P(25)->P(25)^2 : 101->10201 numPrime/numOdd = 1226/5049 = 0.242820 P(26)->P(26)^2 : 103->10609 numPrime/numOdd = 1267/5252 = 0.241241 P(27)->P(27)^2 : 107->11449 numPrime/numOdd = 1353/5670 = 0.238624 P(28)->P(28)^2 : 109->11881 numPrime/numOdd = 1394/5885 = 0.236873 P(29)->P(29)^2 : 113->12769 numPrime/numOdd = 1493/6327 = 0.235973 P(30)->P(30)^2 : 127->16129 numPrime/numOdd = 1846/8000 = 0.230750 P(31)->P(31)^2 : 131->17161 numPrime/numOdd = 1944/8514 = 0.228330 P(32)->P(32)^2 : 137->18769 numPrime/numOdd = 2108/9315 = 0.226302 P(33)->P(33)^2 : 139->19321 numPrime/numOdd = 2156/9590 = 0.224818 P(34)->P(34)^2 : 149->22201 numPrime/numOdd = 2454/11025 = 0.222585 P(35)->P(35)^2 : 151->22801 numPrime/numOdd = 2511/11324 = 0.221741 P(36)->P(36)^2 : 157->24649 numPrime/numOdd = 2692/12245 = 0.219845 P(37)->P(37)^2 : 163->26569 numPrime/numOdd = 2877/13202 = 0.217922 P(38)->P(38)^2 : 167->27889 numPrime/numOdd = 3004/13860 = 0.216739 P(39)->P(39)^2 : 173->29929 numPrime/numOdd = 3201/14877 = 0.215164 P(40)->P(40)^2 : 179->32041 numPrime/numOdd = 3395/15930 = 0.213120 P(41)->P(41)^2 : 181->32761 numPrime/numOdd = 3470/16289 = 0.213027 P(42)->P(42)^2 : 191->36481 numPrime/numOdd = 3825/18144 = 0.210813 P(43)->P(43)^2 : 193->37249 numPrime/numOdd = 3901/18527 = 0.210558 P(44)->P(44)^2 : 197->38809 numPrime/numOdd = 4044/19305 = 0.209479 P(45)->P(45)^2 : 199->39601 numPrime/numOdd = 4118/19700 = 0.209036 P(46)->P(46)^2 : 211->44521 numPrime/numOdd = 4580/22154 = 0.206735 P(47)->P(47)^2 : 223->49729 numPrime/numOdd = 5058/24752 = 0.204347 P(48)->P(48)^2 : 227->51529 numPrime/numOdd = 5225/25650 = 0.203704 P(49)->P(49)^2 : 229->52441 numPrime/numOdd = 5306/26105 = 0.203256 P(50)->P(50)^2 : 233->54289 numPrime/numOdd = 5471/27027 = 0.202427 P(51)->P(51)^2 : 239->57121 numPrime/numOdd = 5740/28440 = 0.201828 P(52)->P(52)^2 : 241->58081 numPrime/numOdd = 5829/28919 = 0.201563 P(53)->P(53)^2 : 251->63001 numPrime/numOdd = 6266/31374 = 0.199720 P(54)->P(54)^2 : 257->66049 numPrime/numOdd = 6540/32895 = 0.198814 P(55)->P(55)^2 : 263->69169 numPrime/numOdd = 6813/34452 = 0.197753 P(56)->P(56)^2 : 269->72361 numPrime/numOdd = 7104/36045 = 0.197087 P(57)->P(57)^2 : 271->73441 numPrime/numOdd = 7194/36584 = 0.196643 P(58)->P(58)^2 : 277->76729 numPrime/numOdd = 7485/38225 = 0.195814 P(59)->P(59)^2 : 281->78961 numPrime/numOdd = 7683/39339 = 0.195302 P(60)->P(60)^2 : 283->80089 numPrime/numOdd = 7781/39902 = 0.195003 P(61)->P(61)^2 : 293->85849 numPrime/numOdd = 8292/42777 = 0.193842 P(62)->P(62)^2 : 307->94249 numPrime/numOdd = 9026/46970 = 0.192165 P(63)->P(63)^2 : 311->96721 numPrime/numOdd = 9245/48204 = 0.191789 P(64)->P(64)^2 : 313->97969 numPrime/numOdd = 9351/48827 = 0.191513 P(65)->P(65)^2 : 317->100489 numPrime/numOdd = 9565/50085 = 0.190975 P(66)->P(66)^2 : 331->109561 numPrime/numOdd = 10348/54614 = 0.189475 P(67)->P(67)^2 : 337->113569 numPrime/numOdd = 10688/56615 = 0.188784 P(68)->P(68)^2 : 347->120409 numPrime/numOdd = 11266/60030 = 0.187673 P(69)->P(69)^2 : 349->121801 numPrime/numOdd = 11390/60725 = 0.187567 P(70)->P(70)^2 : 353->124609 numPrime/numOdd = 11630/62127 = 0.187197 P(71)->P(71)^2 : 359->128881 numPrime/numOdd = 11992/64260 = 0.186617 P(72)->P(72)^2 : 367->134689 numPrime/numOdd = 12480/67160 = 0.185825 P(73)->P(73)^2 : 373->139129 numPrime/numOdd = 12860/69377 = 0.185364 P(74)->P(74)^2 : 379->143641 numPrime/numOdd = 13239/71630 = 0.184825 P(75)->P(75)^2 : 383->146689 numPrime/numOdd = 13490/73152 = 0.184411 P(76)->P(76)^2 : 389->151321 numPrime/numOdd = 13883/75465 = 0.183966 P(77)->P(77)^2 : 397->157609 numPrime/numOdd = 14412/78605 = 0.183347 P(78)->P(78)^2 : 401->160801 numPrime/numOdd = 14673/80199 = 0.182957 P(79)->P(79)^2 : 409->167281 numPrime/numOdd = 15203/83435 = 0.182214 --------------------------------------------------------- EXAMINING THE #P[P(n)->P(n)^2] AND #P_Corrected[P(n)->P(n)^2] FORMULAE #P[P(2)->P(2)^2] = 7; #P_Corrected[P(2)->P(2)^2] = 7; actual number primes in this interval = 6 #P[P(3)->P(3)^2] = 11; #P_Corrected[P(3)->P(3)^2] = 11; actual number primes in this interval = 11 #P[P(4)->P(4)^2] = 25; #P_Corrected[P(4)->P(4)^2] = 24; actual number primes in this interval = 25 #P[P(5)->P(5)^2] = 32; #P_Corrected[P(5)->P(5)^2] = 31; actual number primes in this interval = 33 #P[P(6)->P(6)^2] = 52; #P_Corrected[P(6)->P(6)^2] = 50; actual number primes in this interval = 54 #P[P(7)->P(7)^2] = 62; #P_Corrected[P(7)->P(7)^2] = 58; actual number primes in this interval = 64 #P[P(8)->P(8)^2] = 87; #P_Corrected[P(8)->P(8)^2] = 81; actual number primes in this interval = 90 --------------------------------------------------------- EXAMINING THE #P[P(n-1)^2->P(n)^2] AND #P_Corrected[P(n-1)^2->P(n)^2] FORMULAE #P[P(1)^2->P(2)^2] = 4; #P_Corrected[P(1)^2->P(2)^2] = 4; actual number primes in this interval = 5 #P[P(2)^2->P(3)^2] = 5; #P_Corrected[P(2)^2->P(3)^2] = 5; actual number primes in this interval = 6 #P[P(3)^2->P(4)^2] = 15; #P_Corrected[P(3)^2->P(4)^2] = 14; actual number primes in this interval = 15 #P[P(4)^2->P(5)^2] = 9; #P_Corrected[P(4)^2->P(5)^2] = 8; actual number primes in this interval = 9 #P[P(5)^2->P(6)^2] = 22; #P_Corrected[P(5)^2->P(6)^2] = 20; actual number primes in this interval = 22 #P[P(6)^2->P(7)^2] = 12; #P_Corrected[P(6)^2->P(7)^2] = 8; actual number primes in this interval = 11 --------------------------------------------------------- EXAMINING THE #Twins[P(n)->P(n)^2] AND #Twins_Corrected[P(n)->P(n)^2] FORMULAE #Twins[P(2)->P(2)^2] = 3; #Twins_Corrected[P(2)->P(2)^2] = 3; actual number twins in this interval = 3 #Twins[P(3)->P(3)^2] = 4; #Twins_Corrected[P(3)->P(3)^2] = 4; actual number twins in this interval = 4 #Twins[P(4)->P(4)^2] = 8; #Twins_Corrected[P(4)->P(4)^2] = 8; actual number twins in this interval = 8 #Twins[P(5)->P(5)^2] = 9; #Twins_Corrected[P(5)->P(5)^2] = 9; actual number twins in this interval = 9 #Twins[P(6)->P(6)^2] = 13; #Twins_Corrected[P(6)->P(6)^2] = 13; actual number twins in this interval = 16 #Twins[P(7)->P(7)^2] = 15; #Twins_Corrected[P(7)->P(7)^2] = 14; actual number twins in this interval = 17 #Twins[P(8)->P(8)^2] = 20; #Twins_Corrected[P(8)->P(8)^2] = 19; actual number twins in this interval = 21 --------------------------------------------------------- EXAMINING THE #Triplets[P(n)->P(n)^2] AND #Triplets_Corrected[P(n)->P(n)^2] FORMULAE #Triplets[P(3)->P(3)^2] = 3; #Triplets_Corrected[P(3)->P(3)^2] = 3; actual number triplets in this interval = 4 #Triplets[P(4)->P(4)^2] = 4; #Triplets_Corrected[P(4)->P(4)^2] = 4; actual number triplets in this interval = 5 #Triplets[P(5)->P(5)^2] = 4; #Triplets_Corrected[P(5)->P(5)^2] = 4; actual number triplets in this interval = 5 #Triplets[P(6)->P(6)^2] = 6; #Triplets_Corrected[P(6)->P(6)^2] = 6; actual number triplets in this interval = 6 #Triplets[P(7)->P(7)^2] = 6; #Triplets_Corrected[P(7)->P(7)^2] = 6; actual number triplets in this interval = 8 #Triplets[P(8)->P(8)^2] = 7; #Triplets_Corrected[P(8)->P(8)^2] = 7; actual number triplets in this interval = 8 --------------------------------------------------------- EXAMINING THE #Quadruplets[P(n)->P(n)^2] AND #Quadruplets_Corrected[P(n)->P(n)^2] FORMULAE #Quadruplets[P(3)->P(3)^2] = 1; #Quadruplets_Corrected[P(3)->P(3)^2] = 1; actual number quadruplets in this interval = 2 #Quadruplets[P(4)->P(4)^2] = 2; #Quadruplets_Corrected[P(4)->P(4)^2] = 2; actual number quadruplets in this interval = 3 #Quadruplets[P(5)->P(5)^2] = 1; #Quadruplets_Corrected[P(5)->P(5)^2] = 1; actual number quadruplets in this interval = 2 #Quadruplets[P(6)->P(6)^2] = 2; #Quadruplets_Corrected[P(6)->P(6)^2] = 2; actual number quadruplets in this interval = 3 #Quadruplets[P(7)->P(7)^2] = 2; #Quadruplets_Corrected[P(7)->P(7)^2] = 2; actual number quadruplets in this interval = 2 #Quadruplets[P(8)->P(8)^2] = 2; #Quadruplets_Corrected[P(8)->P(8)^2] = 2; actual number quadruplets in this interval = 2 --------------------------------------------------------- Execution time = 43 seconds.